Automorphisms, Derivations And Second Cohomology Of A Class Of Generalized Witt Algebras

Let W[x±1,e^±x] be the generalized Witt algebra in the one-variable case based onusing the exponential functions. W[x,e^x], W[x,e^±x] and W[x±1,e^x] be its Lie subalgebras.Kawamoto et al. determined the automorphism groups of W[x,e^x] and W[x,e^±x]. In thispaper, we determine the automorphism groups of two other Lie algebras W[x±1,e^x] andW[x±1,e^±x], the derivations of all the four Lie algebras and the second cohomology groupsof W[x,e^x] and W[x,e^±x]. The main results are as follow:Let W be any one of the four Lie algebras, denote by Aut(W) the automorphismgroup of W. Then Aut(W[x±1,e^x])～= F? and Aut(W[x±1,e^±x])～= Z/2Z F⁺. Denote byDer(W) and Inn(W) the spaces of derivations and inner derivations of W, respectively.Then Der(W) = Inn(W)⊕d . Let H2(W,F) be the second cohomology group of W.Then H²(W[x,e^x],F) = H²(W[x,e^±x],F) = 0.